Quantum computing and quantum information - · PDF fileQuantum algorithms Main applications of quantum computing: - Hidden abelian subgroup problem (Shor’s alg., exponential speedup)

Quantum computing and quantum information

KAIS GROUP

Main themes

Quantum algorithms. In particular for quantum chemistry

Entanglement for complex chemical and biological systems

Adiabatic quantum algorithms for quantum simulation and more

arXiv:1502.00671 [physics.bio-ph] arXiv:1408.3556 [physics.bio-ph] arXiv:1204.5262 [physics.chem-ph]

arXiv:1311.2555 [quant-ph] arXiv:1310.1933 [quant-ph]

arXiv:1004.2242 [cs.NE] arXiv:1009.5625 [quant-ph] arXiv:1307.7220 [quant-ph] arXiv:1302.1946 [quant-ph] arXiv:1302.0579 [quant-ph] …

+2 papers in progress

Molecular Hamiltonian Quantum circuit

Adiabatic quantum device (quantum annealer) Hard Optimization problems

Quantum algorithms

Quantum algorithms Main applications of quantum computing:

- Hidden abelian subgroup problem (Shor’s alg., exponential speedup)

- Unstructured search (Grover’s, quadratic speedup)

- Quantum simulation (exponential speedup)

Need thousands of qubits to surpass classical computation arXiv:quant-ph/0205095 e.g. quantum chemistry: exact diagonalization

Classical: maximum 50 ~ 60 orbitals Quantum: need to handle 50 ~ 100 orbitals at least to surpass classical

Need 50 ~ 100 qubits

Quantum simulation is one of the most promising near-term applications for quantum computation that could demonstrate significant advantage over classical algorithms.

Quantum simulation Goal: simulate properties of quantum systems

1) Dynamic properties: time evolution 𝑒−𝑖𝐻𝑡 for a given Hamiltonian H

2) Static properties: for example the ground state of a given H

𝑈 = 𝑒−𝑖𝐻𝑡

Target unitary

𝑈′ = 𝑈𝑚𝑈𝑚−1…𝑈2 𝑈1

Elementary gate sequence

Gate decomposition schemes: Trotter-Suzuki Solovay-Kitaev Taylor Series … Heuristic approach

Gate decomposition

Group leaders optimization algorithm [Daskin, Kais, JCP, 2011; Mol. Phys., 2011] Input: Gate set, target unitary U and error 휀 Output: Gate sequence U’ such that

𝑈 − 𝑈′ ≤ 휀

Quantum simulation of molecular Hamiltonians

Molecular Hamiltonian

𝐻 = ℎ𝑖𝑗𝑎𝑖†𝑎𝑗

𝑖,𝑗

+1

2 ℎ𝑖𝑗𝑘𝑙𝑎𝑖

†𝑎𝑗†

𝑖,𝑗,𝑘,𝑙

𝑎𝑘𝑎𝑙

Second-quantized form

Jordan-Wigner transform

J-W

𝐻 = ℎ𝛼𝑖 𝜎𝛼

𝑖

𝑖,𝛼

+ ℎ𝛼𝛽𝑖𝑗𝜎𝛼𝑖𝜎𝛽

𝑗

𝑖𝑗𝛼𝛽

+⋯

Qubit Hamiltonian (sum of tensor products of Pauli matrices)

𝑈 = 𝑒−𝑖𝐻𝑡

Target unitary

Gate decomposition scheme

Rephrase gate decomposition as an optimization problem:

A. Daskin and S. Kais, Mol. Phys. 109, 761 (2011).

min𝑈′

1 − 𝛼 ∙1

𝑁𝐓𝐫 𝑈′𝑈† +

𝛽

Cost

𝑈 = 𝑒−𝑖𝐻𝑡

Target unitary

𝑈′ = 𝑈𝑚𝑈𝑚−1…𝑈2 𝑈1

Elementary gate sequence

Gate decomposition

𝑈′ is a sequence of gates from a user-defined elementary gate set 1

𝑁𝐓𝐫 𝑈′𝑈† is the correctness of the circuit U’

Cost is a user-defined function that decides the cost of each elementary gate

The heuristic algorithm strive to find a globally optimized gate sequence with respect to the gate set and cost function assigned by the user.

Gate decomposition: numerics

A. Daskin and S. Kais. J. Chem. Phys. 134, 144112 (2011).

Example: H2O molecule

Molecular Hamiltonian

Second-quantized form

J-W Qubit Hamiltonian (sum of tensor products of Pauli matrices)

Target unitary

Quantum circuit

More quantum algorithms…

Quantum circuits for solving linear systems

Quantum algorithm for solving Poisson equation (collaboration with Joseph Traub at Columbia U.)

Quantum algorithm for multiple network alignment

Universal quantum circuit scheme for finding complex eigenvalues

Experimental realization on NMR (collaboration with Jiangfeng Du at Univ. Sci. Tech. China)

arXiv:1110.2232 [quant-ph]

arXiv:1302.1946 [quant-ph]

arXiv:1207.2485 [quant-ph]

arXiv:1307.7220 [quant-ph]

arXiv:1302.0579 [quant-ph]

Quantum circuit schemes

Experimental collaborations

Entanglement and coherence in biological and chemical systems

Avian compass

Coherent energy transfer in photosynthetic systems

Radical pair mechanism for avian compass

External magnetic field B alternates the state of a weakly coupled radical pair

S-T Conversion influenced by Zeeman effect and hyperfine interaction

Inspired applications: synthetic donor-bridge-acceptor compass, chemical compass using magnetic nanostructures

Radical pair mechanism

Radical pair mechanism: numerics

Quantum entanglement measured by Negativity

1. Yields dependent on angle; 2. Entanglement dependent on angle if hyperfine coupling is anisotropic.

Y. Zhang, G. Berman, S. Kais.Int. J. Quant. Chem., 115, 15 (2015)

Entanglement and coherence in FMO and LH2 complex

S. Yeh, J. Zhu, S. Kais, J. Chem. Phys. 137, 084110 (2012); S. Yeh, S. Kais, arXiv:1408.3556v1 (2014)

Model Hamiltonian

ℋ = ℋ𝑆 +ℋ𝐵 +ℋ𝑆𝐵; system, bath, system-bath interaction

ℋ𝑆: 1-exciton and 2-exciton basis

ℋ𝐵: phonon bath

LH2 FMO

Bacteriochlorophyll (BChl)

Numerical simulation Simulation schemes

◦ Redfield equation assumes 𝐻𝑆𝐵 ≪ 𝐻𝑆

◦ Förster theory assumes 𝐻𝑆𝐵 ≫ 𝐻𝑆

◦ Hierarchical equation of motion (HEOM) no requirements

BChl coulomb coupling ≈ electron-bath coupling

Q. Shi, L. P. Chen, G. J. Nan, R. X. Xu, Y. J. Yan, J. Chem. Phys., 130, 084105 (2009) J. Zhu, S. Kais, P. Rebentrost, and A. Aspuru-Guzik, J. Phys. Chem. B, 115, 1531 (2011). J. Zhu, S. Kais, A. Aspuru-Guzik, S. Rodriques, B. Brock, and P. J. Love, arXiv:1202.4519v1 (2012).

Scaled HEOM (Shi et al.)

We apply scaled HEOM on simulation of the dynamics of excitation energy transfer in LH2 and FMO complex

Simulation results: LH2

S1 initially excited

S2 initially excited

S. Yeh, J. Zhu, S. Kais, J. Chem. Phys. 137, 084110 (2012);

Simulation results: FMO

7 sites

+ additional site

S. Yeh, S. Kais, arXiv:1408.3556v1 (2014)

Adiabatic quantum computing

Y. Cao, R. Babbush, J. Biamonte, S. Kais, Phys. Rev. A, 91, 1, 012315, 2015;

Adiabatic quantum computing

𝐻 𝑠 = 1 −𝑠

𝑇𝐻1 +

𝑠

𝑇𝐻2 𝛿 =

1

poly(𝑛) for efficient, accurate computation

𝛿

Transverse Ising Model (TIM) 𝐻 = Δ𝑖𝑋𝑖

𝑖+ ℎ𝑖𝑍𝑖

𝑖+ 𝐽𝑖𝑗𝑍𝑖𝑍𝑗

𝑖,𝑗

Large-scale implementation of TIM ~1000 qubits Initial Hamiltonian: single X fields Final Hamiltonian: ZZ interactions

ℎ𝑖𝑍𝑖𝑖 + 𝐽𝑖𝑗𝑍𝑖𝑍𝑗𝑖,𝑗

Could embed hard optimization problems into the Hamiltonian

Set Cover with Pairs Given two sets of nodes U, S and a graph G(V, E) where 𝑉 = 𝑈 ∪ 𝑆

Find a minimum size subset 𝐴 ⊆ 𝑆 such that every element of U is connected to two elements in A.

Y. Cao, S. Jiang, D. Perouli, S. Kais. <Work in progress>

Example. A network of users U and facilities S. Find the minimum set of facilities such that each user is covered by at least two facilities.

Reduction to Ising Model

Set Cover with Pairs (G, U, S)

Integer programming (G, U, S) + auxiliary variables T

Ising Hamiltonian ℎ𝑖𝑍𝑖𝑖 + 𝐽𝑖𝑗𝑍𝑖𝑍𝑗𝑖,𝑗

Y. Cao, S. Jiang, D. Perouli, S. Kais. <Work in progress>

Ground state 𝜓 Solution

Runtime analysis Consider SCPP instances with different input sets U, S and random G;

For each SCPP instance, construct an Ising Hamiltonian H’;

Start from Δ𝑖𝑋𝑖𝑖 and evolve it adiabatically to H’;

Find the minimum time T needed to yield a final state encoding the solution with probability 0.95.

Adiabatic quantum simulation Goal: simulate properties of quantum systems

1) Dynamic properties: time evolution 𝑒−𝑖𝐻𝑡 for a given Hamiltonian H

2) Static properties: for example the ground state of a given H

Molecular Hamiltonian

Second-quantized form

J-W Qubit Hamiltonian (sum of tensor products of Pauli matrices)

Target unitary

Quantum circuit

Ground state

The qubit Hamiltonian is usually many-body

Adiabatic quantum simulation

Molecular Hamiltonian

Second-quantized form

J-W Qubit Hamiltonian (sum of tensor products of Pauli matrices)

Target unitary

Quantum circuit

Ground state

The qubit Hamiltonian is usually many-body, which is unphysical to realize

Example: H2 in minimal STO-3G basis

Need to reduce the Hamiltonian to 2-body

Reduction by perturbation theory Basic idea:

Given a k-local target Hamiltonian 𝐻targ on n qubits;

Find a 2-local Hamiltonian 𝐻 on n + poly(n) qubits such that

𝜆 𝐻targ − 𝜆(𝐻 ) ≤ 휀

Lowest 2n energy levels

Qubit Hamiltonian 𝐻targ

(sum of tensor products of Pauli matrices)

Reduced 2-local Hamiltonian 𝐻

Y. Cao, R. Babbush, J. Biamonte, S. Kais. Phys. Rev. A, Vol. 91, Issue 1, 012315, 2015

Summary Quantum algorithms. In particular for quantum chemistry

Target unitary evolution Quantum circuit

Adiabatic quantum device (quantum annealer) Set cover with pairs

Gate decomposition

min𝑈′

1 − 𝛼 ∙1

𝑁𝐓𝐫 𝑈′𝑈† +

𝛽

Cost

Exciton transfer

Radical pair mechanism

Electronic structure calculation

Entanglement for complex chemical and biological systems

Adiabatic quantum algorithms for quantum simulation and more

Thanks for your attention!